RUI: Applications of Non-Commutative Algebra to Character Varieties

本课题关注有限生成群到李群表示的模空间。许多有趣的模空间以这种方式出现：表面上的扁平纤维束和（抛物线）希格斯束就是例子。这些空格有时被称为字符变体。特征变的研究与矩阵群积的不变性理论相交叉，因此其结构在很大程度上要归功于一般矩阵（多项式不定数上的矩阵）的非交换环。由于特征变异可以被认为是基空间（拓扑）的基本群到李群（几何）的表示的等价类的空间，它们在微分几何和数学物理中都有应用。本文的中心目标是利用非交换环理论的强关系来回答有关字符变体的同伦、算术和动态结构的问题。例如，数学家对数学对象进行分类的努力与生物学家对物种进行分类的努力并无不同。这个项目的目的是分类和推进对某些数学对象的理解，这些对象被称为“字符品种”。非正式地，字符变体提供了编码等价方式的数据，可以将对象的柔性形状与同一对象的刚性形状相关联。例如，橡皮筋是一个有弹性的“圆”，但一旦规定了半径，就确定了一个刚性的圆。对字符变化几何的研究在某种意义上可以被认为是“几何中的几何”。在某种程度上，由于这个原因，字符的变化在微分几何和数学物理中都有应用。在这个项目中，PI和他的合作者和学生们将使用与人物品种相关的某些非对称结构作为工具，探索和回答有关其形状的新问题，以及它们的元素如何通过某些运动相互关联。这个项目的另一个重要部分是在PI的实验代数和几何实验室培训本科生研究助理，并开展社区外展活动，传播纯数学的重要性和美丽。

This project concerns moduli spaces of representations of finitely generated groups into Lie groups. Many moduli spaces of interest arise in this fashion: flat fiber bundles over a surface, and (parabolic) Higgs bundles are examples. These spaces are sometimes called character varieties. The study of character varieties intersects the invariant theory of products of matrix groups, and so owes much of its structure to the non-commutative ring of generic matrices (matrices over polynomial indeterminates). Since character varieties may be considered spaces of equivalence classes of representations of the fundamental group of a base space (topology) into a Lie group (geometry), they find applications throughout differential geometry and mathematical physics. The central goal of the proposed work is to use the strong relationship to non-commutative ring theory to answer questions about the homotopic, arithmetic, and dynamical structure of character varieties.Mathematicians' endeavor to classify mathematical objects is not unlike, for example, the biologists' endeavor to classify species. The purpose of this project is to classify and advance the understanding of certain mathematical objects called "character varieties." Informally, character varieties provide data which encodes equivalent ways one can associate the flexible shape of a object to a rigid shape of that same object. For example, a rubber band is a flexible "circle," but once one stipulates a radius a rigid circle is determined. The study of the geometry of character varieties in some sense can be thought of as the "geometry of geometry." In part because of this, character varieties find applications throughout differential geometry and in mathematical physics. In this project, the PI and his collaborators and students will use certain non-symmetrical structures associated with character varieties as a tool to explore and answer novel questions about their shape, and how their elements relate to each other by certain motions. Another important part of this project is the training of undergraduate research assistants in the PI's Experimental Algebra and Geometry Lab, and the conducting of community outreach activities which communicate the importance and beauty of pure mathematics.